Problem: Let $h(x)=2\cdot5^x$. Find $h'(x)$. Choose 1 answer: Choose 1 answer: (Choice A) A $\ln(5)\cdot 10^x$ (Choice B) B $2\cdot \ln(x)\cdot 5^x$ (Choice C) C $2\cdot \ln(5)\cdot 5^x$ (Choice D) D $5\cdot 10^{x-1}$
Solution: The expression for $h(x)$ includes an exponential term. Remember that the derivative of the general exponential term $a^x$ (where $a$ is any positive constant) is $\ln(a)\cdot a^x$. Put another way, $\dfrac{d}{dx}(a^x)=\ln(a)\cdot a^x$. $\begin{aligned} h'(x)&=\dfrac{d}{dx}(2\cdot5^x) \\\\ &=2\dfrac{d}{dx}(5^x) \\\\ &=2\cdot\ln(5)\cdot5^x \end{aligned}$ In conclusion, $h'(x)=2\cdot\ln(5)\cdot 5^x$.